Comments Welcome

The Disposition E®ect and Momentum ¤

Mark Grinblatt and Bing Hany

Current Version: April 2, 2002

¤Grinblatt and Han are both from the Anderson School at UCLA. The authors thank Andrew Lo andJiang Wang for providing the MiniCRSP database used in the paper's empirical work, the UCLA AcademicSenate for ¯nancial support, Shlomo Benartzi, Steve Cauley, Bhagwan Chowdhry, Wayne Ferson, Mark Gar-maise, David Hirshleifer, Francis Longsta®, Monika Piazzesi, Richard Roll and Jiang Wang for invaluablediscussions, and seminar participants at Boston College, Emory University, MIT, NYU, Ohio State Univer-sity, Penn State University, UC Berkeley, UC Irvine, UCLA, University of North Carolina at Chapel-Hill,University of Texas at Austin, University of Washington, University of Wisconsin-Madison, and WashingtonUniversity at St. Louis for comments on earlier drafts.

yCorrespondence to: Bing Han, Anderson Graduate School of Management, UCLA; 110 Westwood Plaza;Los Angeles; CA 90095-1481. Phone: (310) 825-8160. E-mail: bhan@anderson.ucla.edu.

Abstract

The Disposition E®ect and Momentum

Prior experimental and empirical research documents that many investors have a lowerpropensity to sell those stocks on which they have a capital loss. This behavioralphenomenon, known as \the disposition e®ect," has implications for equilibrium prices.We investigate the temporal pattern of stock prices in an equilibrium that aggregatesthe demand functions of both rational and disposition investors. The disposition e®ectcreates a spread between a stock's fundamental value { the stock price that wouldexist in the absence of a disposition e®ect { and its market price. Even when a stock'sfundamental value follows a random walk, and thus is unpredictable, its equilibriumprice will tend to underreact to information. Spread convergence, arising from therandom evolution of fundamental values and updating of the reference prices, generatespredictable equilibrium prices. This convergence implies that stocks with large pastprice run-ups and stocks on which most investors experienced capital gains have higherexpected returns than those that have experienced large declines and capital losses.The pro¯tability of a momentum strategy, which makes use of this spread, dependson the path of past stock prices. Cross-sectional empirical tests of the model ¯ndthat stocks with large aggregate unrealized capital gains tend to have higher expectedreturns than stocks with large aggregate unrealized capital losses and that this capitalgains \overhang" appears to be the key variable that generates the pro¯tability of amomentum strategy. When this capital gains variable is used as a regressor along withpast returns and volume to predict future returns, the momentum e®ect disappears.

Momentum, which is the persistence in the returns of stocks over horizons between three

months and one year, remains one of the most puzzling anomalies in ¯nance. Jegadeesh and

Titman (1993), for example, found that past winning stocks, as measured by returns over

the prior six months tended to subsequently outperform past losing stocks by about twelve

percent per year. Various explanations for momentum have been advanced, but few have

stood up to rigorous empirical tests or calibrations. Moreover, most of these explanations

are more rooted in attempts to explain momentum than in developing general theories of

how demand for stocks a®ects equilibrium pricing.

Momentum-motivated theories have often been categorized as belonging to the ¯eld of

behavioral ¯nance, there is a seemingly unrelated branch of the literature in behavioral

¯nance that documents stylized facts about investor behavior. It is motivated by experiments

on individual investors and observations of their behavior in real ¯nancial markets. It rarely

attempts to link its stylized facts about investor behavior to equilibrium prices. Perhaps the

most well-documented behavioral regularity in this literature is what Shefrin and Statman

(1985) termed \the disposition e®ect." This is the tendency of investors to hold onto their

losing stocks to a greater extent than they hold onto their winners.1

This paper analyzes how aggregate demand and equilibrium prices evolve over time when

they are a®ected by the existence of a ¯xed proportion of investors who exhibit the dispo-

sition e®ect. It is possible to analytically prove that if some investors are subject to the

disposition e®ect, then stocks with aggregate unrealized capital gains tend to outperform

stocks with aggregate unrealized capital losses. This paper shows, both theoretically and

empirically, that the disposition e®ect may account for the tendency of past winning stocks

to subsequently outperform past losing stocks.

The intuition for our model is rather simple. Assume for the moment that some investors,

the disposition investors, perturb otherwise rational demand functions for a stock they own

because they have experienced unrealized capital gains or losses in the stock. Such investors

would tend to have higher demand for losing stocks than for winning stocks, other things

being equal. If demand for that same stock by other investors is not perfectly elastic,

then such a demand perturbation, induced by a disposition e®ect, tends to generate price

underreaction to public information. Stocks that have been privy to good news in the

past would have excess selling pressure relative to stocks that have been privy to adverse

1The disposition e®ect is sometimes linked to, but is really distinct from loss aversion. Loss aversionoccurs when two conditions are met: (i) the decline in utility for a loss (measured relative to a referencepoint) exceeds the increase in utility for an equal sized gain (relative to the same reference point); (ii) agamble that is always a loss relative to the reference point generates higher utility than a certain loss with thesame mean while the reverse preference applies to gambles that are always gains. In a market with a singlerisky asset, one can see how an investor might be more likely to substitute a risk-free asset for a risky assetin a gain situation (where utility is locally concave) than a loss situation (where utility is locally convex).However, in a multiasset multiperiod framework, it is necessary to argue that reference prices change toinduce substitution of one risky asset with a paper gain or loss for another. This complex setting is atypicalin the modeling of loss aversion.

1

information. This produces a spread between the fundamental value of the stock { its

equilibrium price in the absence of any disposition e®ect among investors { and the market

price of the stock. Aggregate investor demand equals supply in the model when the stock

price is a weighted average of its fundamental value and a reference price (related to the

basis at which disposition investors acquired the stock).

The demand perturbation and associated spread does not, per se, generate momentum

in stock returns. For momentum to arise, there must be some mechanism that forces market

prices to revert to fundamental values. As one example of the importance of this reversion,

note that if all investors are identical and subject to the disposition e®ect, market prices

deviate from fundamental values. However, in the absence of trading, it would be reasonable

to conclude that the reference price (such as the cost basis) of disposition investors would

not change. In this case, the associated demand perturbation generates a market price that

is a constant weighted average of the stock's fundamental value and a ¯xed reference price.

Because the weights are constant, changes in market prices are just dampened versions of

changes in fundamental values. Hence, if fundamental values follow a random walk, so do

market prices, despite their underreaction to information about fundamentals.

One could imagine various forces that drive market prices to their fundamental values,

thus generating momentum. A press release about an acquisition, an earnings announcement,

and a reduction in uncertainty all are events that may make some investors rely less on

their behavioral tendency to sell winners and hold onto losers. However, it is surprising to

discover that a random process for a stock's fundamental value, in and of itself, makes a

stock's reference price, and hence its market price, catch up with its associated fundamental

value. In other words, trading, which arises in the model only because some investors exhibit

the disposition e®ect, produces mean reversion in the spread between a stock's fundamental

value and its market price. By de¯nition, the stochastic process for a stock's market price

is the stochastic process for its fundamental value less that for its spread; hence, any mean

reversion in the stochastic process for the spread implies that stocks with positive spreads

have risk-adjusted mean returns that exceed the risk-free return, while those with negative

spreads have risk-adjusted mean returns that are below the risk-free return. The mean return

of a stock that is a big winner, which tends to have a positive spread between its fundamental

value and market price, thus tends to exceed that of a loser, which tends to have a negative

spread.

The di®erence between the market price and the reference price is proportional to the

spread between the fundamental value and the market price. Thus, it, too, is of the same

sign as the expected return. If the reference price is related to some aggregate cost basis

in the market, as the motivation for this paper suggests, this di®erence is a gain measured

relative to a reference price that is the cost basis.

The empirical implications of the model, outlined above, are veri¯ed with cross-sectional

\Fama-MacBeth" regressions that make use of a \capital gains overhang" regressor to proxy

2

for aggregate unrealized capital gains or losses. In all of our regression speci¯cations, the

gain variable predicts future returns, even after controlling for the e®ect of past returns,

but the reverse is rarely true. Indeed, in most of our regression speci¯cations, there is no

momentum e®ect once the disposition e®ect is controlled for with the gain regressor.

Section I of the paper presents results derived from a model with two types of investors:

One type has no disposition e®ect and is fully rational; the other is identical to the ¯rst type

except that the disposition e®ect perturbs his demand function for stocks. These include

results about the temporal pattern of equilibrium prices conditional on both aggregate capital

gains and past returns. It also includes results about the determinants of trades and volume.

Section II presents empirical data and provides numerous tests illustrating that our ¯ndings

are not due to omitted variables that others have used in the literature to analyze momentum.

Our main ¯nding here is that the capital gains overhang is a critical variable in any study

of the relation between past returns and future returns, as the theory predicts. Section III

discusses the relation of our work to prior literature, both theoretical and empirical. Section

IV concludes the paper.

I. The Model

In an attempt to make the model as simple and analytically tractable as possible, we

focus on how the partial demand function of a single risky stock among possibly many assets

has its equilibrium price path a®ected by the disposition e®ect. We assume

² The economy contains a riskless asset in perfectly elastic supply and with a zero rateof return

² The risky stock that we focus on is in ¯xed supply normalized to one unit. Public newsabout the fundamental value of the stock arrives at discrete dates t = 0; 1; 2; : : : just

prior to trading on those dates

² There are two types of investors:{ The fraction 1¡ ¹ are type-r \rational" investors{ The fraction ¹ are type-d \disposition" investors

² Date t demand functions per unit of each investor-type's mass in the market are re-spectively given by

Drt = 1 + bt(Ft ¡ Pt) (1)

Ddt = 1 + bt[(Ft ¡ Pt) + ¸(Rt ¡ Pt)] (2)

where

3

{ Ft is the stock's fundamental value2

{ Pt is the price of the stock

{ bt is a positive parameter that represents the slope of the rational component of

the demand functions for the stock.3 This parameter is actually a function that

can be contingent on just about anything in the date t information set of investors,

including Pt, Ft, Rt, ¸, ¹, historical values of these variables, or corresponding

information for other assets4

{ Rt is a reference price relative to which type-d investors measure their gains or

losses. It can be thought of as the average cost basis per share of type-d investors'

holdings at the begining of period t (so it is known after trading at period t¡ 1){ ¸ is a positive constant parameter measuring the relative importance of the dis-

position component of demand for type-d investors

For ¸ > 0, one of the two investor-types is relatively more averse to realizing losses.5

When this investor-type has a paper capital loss, he holds more shares than his rational

counterpart. It would be wrong to model him as focused only on stocks that have declined

in value since their purchase. Disposition investors are not characterized by a propensity

2For our results, it is not necessary to be precise about how one arrives at this fundamental value. Wewould merely like the fundamental value to converge to the rational equilibrium price as the number ofdisposition agents converges to zero. Two speci¯c alternatives come to mind. In the ¯rst, the fundamentalvalue at date t is the market price that would prevail at date t if all agents were fully rational but behavedunder the assumption that, in the future there would be disposition agents and equilibria as speci¯ed inthe model. This interpretation of F makes the fundamental value a function of ¸. In the appendix, weanalytically compute such fundamental values and market prices for a multiperiod exponential model. Inthe second, the date t fundamental value is the price that would prevail if all agents were fully rationaland assumes that, in the future, all agents continue to be fully rational. The fundamental value is thus thepresent value of the free cash °ow stream of the stock conditional on all information currently available,adjusted for the risk premium.

3To isolate the impact of the disposition e®ect, and avoid possible confounding e®ects arising, for example,from di®erences in risk aversion across investor-types, we assume that all investor-types have the same slopeto the rational component of their demand functions, bt.

4We can think of bt as being whatever solves for the optimal demand function given a utility function.The fact that demand functions are not perfectly elastic re°ects some risk aversion, capital constraint,or other force restraining unlimited trade by investors. In the interest of lucidity, we see no point toarti¯cially complicate the model with utility functions. The solution to rational investor demand may a®ectthe fundamental value; beyond this, however, it is not present in the equation that determines the equilibriumprice.

5We do not include capital gain tax in our model. The disposition e®ect is more puzzling in such anenviornment. When short term gains (losses) are taxed as ordinary income and long term gains (losses) aretaxed at a lower rate, Constantinides (1983) ¯nds generally it is optimal to realize loss earlier and delayselling winners (just the opposite of disposition e®ect). Klein (2001) analyzes this capital lock-in e®ect andargues that it implies stock return reversal over long horizon. In our framework, adding capital gain tax isequivalent to the case where ¸ < 0 (but for the rational investors instead). Everything proceeds exactly thesame as follows. In this case, a contrarian (rather than momentum) strategy is pro¯table.

4

to accumulate more stock than rational behavior would suggest. Rather, the experimental

and empirical evidence only indicates that their propensity to sell stocks they have lost

money on is less than their propensity to sell stocks they have made money on. Hence, the

other side of the coin of loss realization aversion is a relatively greater propensity to sell

stocks experiencing a gain. As our demand function suggests, when good news about the

fundamental of the stock arrives after the loss realization averse investor has bought it, he

experiences a paper gain, and sells more of the stock than he would if he were fully rational

and the stock was trading at the same price.

The experimental and empirical evidence focuses on active buying and selling behavior.

Modeling equilibrium requires that buying and selling behavior be endogenously derived

from demand functions. For an investor to have a greater propensity to sell a stock with a

paper gain, the excess demand of that investor must be lower for such a stock than it would

be in the absence of disposition behavior (as described above). For an investor to have a

lower propensity to sell a stock with a paper loss, the excess demand of that investor must

be higher for that stock. In the interest of parsimony and tractability, equation (2) assumes

that percentage demand deviation from rational behavior is proportional to the capital gain

or loss overhang where the deviation parameter ¸ is constant.6

Note that if ¸ is zero, (or, alternatively, if ¹ = 0), so that all investors are \rational,"

aggregate demand, a weighted average of these two demand functions is

(1¡ ¹)Drt + ¹Ddt = 1 + bt(Ft ¡ Pt)

and thus, with bt 6= 0, supply equals demand when Ft = Pt. In this sense, the fundamentalvalue F is simply the price that would prevail in the absence of a disposition e®ect. One

of the main goals of this paper is to assess how the disposition e®ect alters the stochastic

process for equilibrium prices. As a benchmark, and consistent with an extensive literature

in ¯nance, we assume that the fundamental value follows a random walk:

Ft+1 = Ft + ²t+1 (3)

where the ²s are i.i.d. and mean zero.7

6Grinblatt and Keloharju (2001), however, estimate that large positive (negative) price deviations fromthe cost basis generate more (fewer) sales.

7The absence of a random walk for fundamental values does not alter any of our results if we interpretall expectations in the paper as risk-neutral expectations. Alternatively, if a random walk does not applyto fundamental values, our results are simply measuring the incremental stochastic processes arising fromdisposition agents. That is, we are providing closed form solutions for how the stochastic processes forequilibrium prices deviates from the stochastic processes that would prevail in the absence of dispositionagents.

5

A. Equilibrium

When ¸ = 0, both investor types are rational and hold 1¡ ¹ and ¹ shares, respectively.Thus, there is no trading. However, when ¸ is nonzero, the demand of the type-d investors

is also a®ected by the unrealized capital gain as measured relative to their reference price.

This provides the key motive for trade between the two types of investors. Thus, at each

date t, aggregate demand equals aggregate supply when

1 + bt(Ft ¡ Pt) + ¹¸bt(Rt ¡ Pt) = 1 (4)

and date t volume, the number of shares changing hands between the two investor-types, is

Vt = ¹jDdt ¡Ddt¡1j

computed at the equilibrium prices for dates t and t ¡ 1. Volume and turnover ratio areidentical here since there is one share outstanding.

The market clearing condition, equation (4), is equivalent to

Pt = wFt + (1¡ w)Rt; where w = 1

1 + ¹¸(5)

demonstrating that the equilibrium market price is a weighted average of the fundamental

value and the reference price. Since 0 < w < 1, the market price underreacts to public infor-

mation about the fundamental value, holding the reference price constant. (Market prices

also underreact for reference prices that follow equilibrium paths, as we will see shortly.) The

degree of underreaction, measured by w, depends on the proportion of disposition investors,

¹, and the relative intensity of the demand perturbation induced by the disposition e®ect,

¸. The fewer the number of disposition investors, and the smaller the degree to which each

perturbs demand, the closer the market price will be to its fundamental value.

To illustrate how changes in the fundamental value of the stock a®ect equilibrium prices,

consider a case where type-d investors' entire holding of shares was purchased at a funda-

mental, market, and reference price of $100 per share last period. Because the three prices

were equal, type-d holdings equal ¹ shares. Suddenly, bad news arrives and the fundamental

value drops to $80. The equilibrium price must end up between the fundamental value of

$80 and the reference price of $100. At a price at or below $80, the stock is too attractive as

the demand functions of both types of investors suggest that they would want to hold more

than they currently own. Equilibrium prices cannot be those at which aggregate demand

for the stock exceeds one share. Similarly, the equilibrium price cannot be at or above $100

either, since aggregate demand would be below one share and both investor-types would like

to sell stock at such prices.

At some price between $80 and $100, the type-r investors exactly accommodate the extra

buying pressure from the type-d investors. The market will be in equilibrium at this price,

6

a weighted average of the fundamental value and the reference price. Since the equilibrium

price must lie between $80 and $100 per share, the downward price response to the news is

sluggish, settling at a weighted average of the past price at which shares were purchased by

disposition investors and the fundamental value. (A similar argument applies for good news

that generates a fundamental value above $100.)

In the illustration above, we had some trades at $100 per share (by assumption) at what

we will refer to as date 0 and then another set of trades occurring at some price between $80

and $100 at date 1. Knowing the parameters of the model would have allowed us to solve

for the date 1 equilibrium price as the single unknown of a linear equation. Assume for the

moment that this price turned out to be $85. To solve for the date 2 equilibrium price as

a function of the date 2 fundamental value, it is critical that we know the date 2 reference

price. It seems reasonable to think that this reference price is going to be some weighted

average of $100 and $85. If we know this reference price, solving for the equilibrium price as

a function of the date 2 fundamental value is again trivial.

B. Reference Price Dynamics

The reference price is established when an investor ¯rst enters into a position and it is

updated as the investor trades. Our speci¯cation for its dynamics here, which is consistent

with the existing experimental and empirical evidence, is that the type-d investors' reference

price satis¯es the di®erence equation 8

Rt+1 = ºtPt + (1¡ ºt)Rt (6)

where ºt, a function of the type-d investors' date t information, lies between 0 and 1.

This speci¯cation implies that the reference price can be any weighted average of past

prices9

Rt+1 =X¿¸0

!t¡¿Pt¡¿ ;X¿

!¿ = 1; !¿ > 0; (7)

For economic content, such as ties to a cost basis, it is useful to think of ºt as linked to the

trading volume of type-d investors. However, our theoretical results are su±ciently general

as to not require even this restriction. For the theoretical ¯ndings in this section, the interval

restriction for ºt should hold in any reasonable speci¯cation of the reference price dynamics.

For example, the reference price dynamics speci¯ed in equation (6) is not only consistent

with the type-d investors' cost basis as the reference price; it is also consistent with the

8More generally, one can introduce a white noise term in this reference price updating equation. Withslight modi¯cation of notation (e.g. in equation (11), replace Rt+1 ¡Rt by Et[Rt+1 ¡Rt], all the followingpropositions continue to hold in the general case (except Proposition 6, which will continue to hold if thewhite noise term in the reference price updating is uncorrelated with innovation in the fundamental value.)

9Iteratively expanding equation (6) implies that the weight on date t ¡ ¿ 's price, !t¡¿ = (1 ¡ ºt)(1 ¡ºt¡1) ¢ ¢ ¢ (1¡ ºt¡¿+1)ºt¡¿ . On the other hand, equation (7) is consistent with ºt¡¿ = !t¡¿

1¡!t¡!t¡1¡¢¢¢¡!t¡¿+1 .

7

reference price being a weighted average of past prices with weights proportional to trade

size, to some function of time proximity of trades in the past, or to any combination of the

two.

Empirical analysis, presented later in the paper, requires a more precise speci¯cation of

ºt. As one example, we might assume that each share is as likely to trade as any other and

that the probability of a trade in a share on any one day is independent of when it has traded

in the past. In this case, a proxy for the aggregate cost basis of the outstanding shares is

obtained by setting ºt as date t turnover, Vt. The associated expression for !t¡¿ , the weighton date t¡ ¿ 's price, is the product of the probabilities that a share did not trade betweendates t¡ ¿ + 1 and t, (1¡ Vt¡¿+1) ¢ ¢ ¢ (1¡ Vt¡1) times the probability that it traded on datet¡ ¿ , Vt¡¿ . The resulting reference price is the expected cost basis of an outstanding shareunder the assumptions given above.

Earlier, we argued that market prices, Pt, respond sluggishly to changes in the fundamen-

tal value, ceteris paribus. Because ºt lies between 0 and 1, the reference price also responds

sluggishly to changes in either the market price or the fundamental value. Substituting

equation (5) into (6), we obtain

Rt+1 = wºtFt + (1¡ wºt)Rt (8)

This equation also points out that the reference price is always reverting to the fundamental

value.

C. Can the Equilibrium Degenerate?

The previous subsections argued that we can solve for an equilibrium in closed form, as

functions of fully rational prices, the fundamental values. However, the fundamental values

cannot be solved for directly, except in very special cases. Moreover, fully rational behavior

recognizes that disposition investors exist. Rest assured that with virtually any reasonable

intertemporal utility function, this does not generate an in¯nite loop that fails to lead to an

equilibrium. The type-r investors, unless they are risk neutral, do not fully undo the impact

of the type-d investors perturbation. As a consequence, the equilibrium does not collapse.

To illustrate this point { that the type-r investors can solve for optimal demand recognizing

the impact of type-d investors who partly mimic them, and that a Walrasian auctioneer

can solve for prices that arise from the speci¯ed demand structure { we derive closed form

solutions for demand and prices in one very special case: In this case, we solve for a date 0

equilibrium involving type-r agents who maximize the expected negative exponential utility

of date 2 wealth, and trade with type-d agents at dates 0 and 1. We also assume that the

fundamental value at date 2 is normally distributed, and that the information about the date

2 value that arrives at date 1 also is normally distributed. The closed form solutions for the

date 0 and date 1 demand functions of type-r investors, along with the date 0 and date 1

8

equilibrium price and fundamental value functions, are found in the appendix. This should

be su±ciently convincing that there is nothing aberrational going on within our model,

despite an underlying complexity that has been deliberately masked.

D. Expected Price Changes

Although the fundamental value, by assumption, follows a random walk, changes in

equilibrium prices are predictable in this model. To see this, note that by equation (5), the

price change can be expressed as:

Pt+1 ¡ Pt = w(Ft+1 ¡ Ft) + (1¡ w)(Rt+1 ¡Rt) (9)

and thus the expected price change is

Et[Pt+1 ¡ Pt] = (1¡ w)(Rt+1 ¡Rt) (10)

Subtracting equation (10) from (9) and substituting in (3) implies

Pt+1 ¡ Pt ¡ Et[Pt+1 ¡ Pt] = w²t+1Since w < 1, this shows that price changes underreact to news about fundamentals.

Equation (10) implies that changes in equilibrium prices are predictable. Reference prices

for the date t+ 1 equilibrium are known at date t. Moreover, from equation (8),

Rt+1 ¡Rt = wºt(Ft ¡Rt) (11)

which, substituted into equation (10), implies

Et[Pt+1 ¡ Pt] = w(1¡ w)ºt(Ft ¡Rt) (12)

Alternatively, this equation is equivalent to

Et[Pt+1 ¡ Pt] = wºt(Ft ¡ Pt) (13)

and to

Et[Pt+1 ¡ Pt] = (1¡ w)ºt(Pt ¡Rt) (14)

by the equilibrium pricing condition, equation (5). These ¯ndings are summarized below:

Proposition 1 If the date t fundamental value exceeds date t's market price (or referenceprice), or if the date t market price exceeds date t's reference price, then the stock price isexpected to increase next period. Similarly, if the date t fundamental value is exceeded bydate t's market price (or reference price), or if the date t market price is exceeded by date t'sreference price, then the stock price is expected to decrease next period. The expected pricechange is proportional to the change in the reference price. Moreover, the expected return isincreasing in the di®erence between the fundamental value and the market (or the reference)price, or in the di®erence between the market and reference prices, and in the weight placedon the current market price in updating the reference price.

9

Hence the date t unrealized gain gt, de¯ned as the di®erence between the market and

reference prices,

gt = Pt ¡Rt (15)

or, alternatively, the date t spread between the fundamental value and the equilibrium price

st = Ft ¡ Pt = 1¡ ww

gt (16)

or the date t gap between the fundamental value and the reference price determine (along

with the weights w and ºt) the expected price change from t to t+1. All of these are driven

by the innovation in the fundamental value, which is the model's only source of uncertainty.

The sign of the date t spread between the fundamental value and the equilibrium price is

the same as the sign of the expected future price change, Et[Pt+1 ¡ Pt], because the formerspread is mean reverting and by de¯nition,

Pt+1 ¡ Pt = (Ft ¡ Pt)¡ (Ft+1 ¡ Pt+1) + (Ft+1 ¡ Ft)

which has an expectation of

Et[Pt+1 ¡ Pt] = (Ft ¡ Pt)¡ Et[Ft+1 ¡ Pt+1] (17)

since the fundamental value follows a random walk. However, with mean reversion in the

spread, the expectation on the right side of equation (17) is smaller in absolute terms than

the term in parentheses it is subtracted from. Hence, if the spread, Ft ¡ Pt, is positive, thedi®erence between the spread and the expected spread is positive; if the spread is negative,

the di®erence is negative. To understand why this mean reversion exists, we prove the

following proposition.

Proposition 2 The gain gt, which is the di®erence between the market and reference prices,mean reverts towards zero. The rate of mean reversion is greater the larger is the absolutemagnitude of ºt, and the smaller is ¸ and ¹. The same result applies to the spread, Ft¡Pt,and to the gap, Ft ¡Rt.

Proof: By equations (3), (11), and (15),

gt+1 ¡ gt = w²t+1 ¡ wºtgt (18)

Hence, Et[gt+1 ¡ gt] = ¡wºtgt. The mean reversion speed is not constant, but is increasingin ºt and w. The spread and gap are constant proportions of the gain so this result also

applies to the spread or gap. ¥

Proposition 2 states that the reference price is always trying to catch up to the latest

fundamental value (and market price). As trading occurs, the reference price gets updated

10

with some weighting of the latest market price, which, in turn, is updated by the latest

fundamental value. This implies that the gain, the spread, and the gap tend to narrow over

time. As the next result indicates, these will widen or change sign only as a consequence of

extraordinary innovations in the fundamental value.

Proposition 3 If the date t gain, gt, is positive (negative), it will continue to be positive(negative) unless a su±ciently large negative (positive) shock to the fundamental arrives.Assume that ºt is positive. Then, the absolute magnitude of the gain will increase nextperiod only if a signi¯cant shock of the same sign as the gain arrives. Otherwise, the absolutemagnitude of the gain will decrease next period. The same results apply to the spread andthe gap.

Proof: All statements follow from equation (18) which states that gt+1 = w²t+1+(1¡wºt)gt.Assume gt > 0. Since 0 < ºt < 1 and 0 < w < 1, gt+1 > 0 unless ²t+1 is su±ciently negative.

Equation (18) also implies gt+1 ¡ gt < 0 if ²t+1 < 0 or if ²t+1 is positive but not too large.An analogous result applies to the negative gain. Since the spread st and gap are constant

multiples of the gain gt, the same conclusions necessarily apply to them. ¥

The results above, applied to the unrealized gain gt, describe expected future price

changes that are conditional on the relation between the market price and the reference

price. This suggests there is an interesting line of empirical work that explores the rela-

tionship between aggregate capital gains and the cross-section of expected returns. It also

may explain why momentum strategies are pro¯table, as stocks with large positive di®er-

ences between the market price and reference price tend to be winning stocks and vice versa.

However, this di®erence is path dependent. There are historical price paths for a stock,

when combined with reasonable speci¯cations for the reference price updating parameter,

ºt, that generate negative gains, and hence negative expected returns, even when past prices

have increased. Similar, anomalous paths exist for losing stocks. Thus, while momentum in

stock returns may be an artifact of the disposition e®ect because past returns are correlated

with variables like aggregate capital gains, our model implies that for a given past return,

some types of paths will generate higher expected returns than others. Lacking a speci¯c

functional form for ºt, it is di±cult to quantify which paths have higher expected future

returns than others. However, it is fair to say that past returns are merely noisy proxies for

behavioral variables, like capital gains, and are likely to be poorer predictors of expected

returns than capital gains proxies if our model is an accurate portrayal of how demand for

stock is generated.

There is one class of past return paths for which the gain and spread are necessarily of

the same sign as the past return. Stocks that have reached a new high (or low) relative to

a reasonably lengthy historical period are those for which past returns, irrespective of the

past return horizon in the historical period are all positive (or negative). Such \consistent

winning" (or \consistent losing") stocks necessarily have investors who acquired the stock

11

at a basis below the current price { thus experiencing a capital gain (or, in the case of the

consistent losers, a capital loss). Given the reference price updating rule, and Proposition 1,

the following result must hold:

Proposition 4 When the market price is at a new high (new low), the gain is positive (neg-ative) and over the next period, the expected change in the market price is positive (negative).

Although there are occasional historical paths for winning stocks that generate negative

spreads and negative expected future returns, intuition tells us that it is unlikely, particularly

if the past return is large, that these paths could dominate the abundant number of paths

for which the spread is positive (and thus, for example, most investors experience capital

gains). This intuition is indeed correct.

Before demonstrating this formally, it is useful to ¯rst express the reference price explicitly

as a weighted average of the fundamental values at previous dates. This is done by applying

equation (11) iteratively.

Rt = wºt¡1Ft¡1 + (1¡ wºt¡1)wºt¡2Ft¡2 + ¢ ¢ ¢+ (1¡ wºt¡1) ¢ ¢ ¢ (1¡ wºt¡n¡1)wºt¡nFt¡n+(1¡ wºt¡1) ¢ ¢ ¢ (1¡ wºt¡n¡1)(1¡ wºt¡n)Rt¡n (19)

This is a complex expression, with coe±cients on the F s and Rt¡n that may be pathdependent. But notice that these nonnegative coe±cients sum to one (which is not surprising

given that the reference price is a weighted average of the current price and the prior period's

reference price and the current price is a weighted average of the current fundamental value

and the current reference price). More recent prices have more in°uence on the current

reference price. As we push further back into history (larger n), we can see that what

happens at a remote historic date matters little for the current reference price. In this sense,

the market is slowly forgetful. The larger the ºs or w, the faster the market forgets. For this

reason, any insights from this model do not depend on initial conditions, provided that the

security under study is su±ciently long-lived.

The low weight placed on the distant past is not the only justi¯cation for our assertion

that initial conditions are unimportant. We can also justify initial conditions with no spread

as a good approximation because the fundamental value and the market price tend to revisit

each other as time evolves. As the following proposition proves, in continuous time, this

occurs with probability one.

Proposition 5 Assume that F follows a di®usion process in continuous time and tradingoccurs continuously. Given any date t spread, st, with probability 1, there is a date in thefuture when the market price equals the fundamental value (a spread and gain of zero).10

10An analogous result holds as an approximation in discrete time with high frequency trading.

12

Proof: Suppose the gain gt is negative. Let F¤ be the all time high of the fundamental value

up until date t. A basic property of Brownian motion is with probability 1, it will eventually

hit any number. Hence with probability 1, there exists a future date ¿ when F¿ = 2F¤ for

the ¯rst time. That means F has a new high at ¿ , and hence by Proposition 4, g¿ > 0. Since

the reference price is a weighted average of historical F 's and F has a continuous sample

path, gt will also have continuous sample path. Hence, by the intermediate value theorem,

sometime between t and ¿ there will be a date when the gain is zero. The proof is the same

when the gain is positive. ¥

We now derive a closed form solution that quanti¯es momentum in stock returns. Using

equation (19), the gain gt = Pt ¡Rt = w(Ft ¡Rt) can be written as:gt = w²t + w(1¡ wºt¡1)²t¡1 + ¢ ¢ ¢+ w(1¡ wºt¡1) ¢ ¢ ¢ (1¡ wºt¡n+1)²t¡n+1

+w(1¡ wºt¡1) ¢ ¢ ¢ (1¡ wºt¡n)(Ft¡n ¡Rt¡n) (20)

This relation turns out to be useful in analyzing how past changes in the fundamental value

a®ect future expected returns. The next proposition actually provides a closed form solution

for this conditional expectation.

Proposition 6 Given a historical horizon of n periods

E[Pt+1 ¡ PtjFt ¡ Ft¡n = x]= w(1¡ w)x

nE[ºt(1 + (1¡ wºt¡1) + : : :+ (1¡ wºt¡1) ¢ ¢ ¢ (1¡ wºt¡n+1)]

+w(1¡ w)E[ºt(1¡ wºt¡1) ¢ ¢ ¢ (1¡ wºt¡n)(Ft¡n ¡Rt¡n)] (21)

This conditional expectation is increasing in x, ceteris paribus, and for positive (negative)x, is positive (negative) if either the absolute magnitude of x or n is su±ciently large. Itis positive (negative) for any positive (negative) x provided that Ft¡n ¡ Rt¡n ¸ 0 (· 0) ornegative (positive) but of su±ciently small absolute magnitude.

Proof: By the law of iterated expectations and equation (12),

E[Pt+1 ¡ PtjFt ¡ Ft¡n = x] = (1¡ w)E[ºtgtjFt ¡ Ft¡n = x]Since the innovation in F is i.i.d.,

E[Ft ¡ Ft¡ijFt ¡ Ft¡n = x] = i

nx

Using this equation, the law of iterated expectations, and equation (20), we obtain

E[ºtgtjFt ¡ Ft¡n = x]= E (E[ºtgtjFt ¡ Ft¡n = x; ºt; ºt¡1; : : : ; ºt¡n])= E (ºtE[gtjFt ¡ Ft¡n; ºt; ºt¡1; : : : ; ºt¡n])= w

x

nE[ºt(1 + (1¡ wºt¡1) + : : :+ (1¡ wºt¡1) ¢ ¢ ¢ (1¡ wºt¡n+1)]

+wE[ºt(1¡ wºt¡1) ¢ ¢ ¢ (1¡ wºt¡n)(Ft¡n ¡Rt¡n)]

13

The sign conclusions are obvious given that the wº terms are between 0 and 1 and because

this implies that the maximum absolute value of the ¯nal expectation term is jE[Ft¡n¡Rt¡n]j,which is bounded. ¥The expectations in Proposition 6 are those of an econometrician who has a prior dis-

tribution for the initial conditions at date 0. However, we have been overly generous in our

bound on the absolute value of the last term in the proof. For virtually any reasonable set

of date 0 values for the fundamental value and the reference price, the ¯nal term within the

expectation is likely to be negligibly small. Hence, from the perspective of an investor at any

date between dates 0 and t ¡ n, who is aware of the initial conditions, and the path takenby the fundamental value, the expected price change between dates t and t + 1 will have

the properties described in Proposition 6. By date t, however, and depending on the initial

conditions, there are occasional, albeit rare paths, for which the gain is negative despite an

increase in the fundamental value between dates t¡ n and t. For this reason, past changesin the fundamental value, and to a greater extent past changes in prices, should be noisier

predictors of future price changes than proxies for the unrealized gains.11 We explore this

issue in the empirical section of the paper.

E. A Back of the Envelope Calculation of the Expected Price Change

To assess whether our model generates expected price changes that bear any resemblance

to those observed by empiricists, we now undertake a back of the envelope calculation of

E[Pt+1 ¡ PtjFt ¡ Ft¡n = x].Assume for simplicity that for all ¿ , º¿ = º. By equations (12) and (20),

E[Pt+1 ¡ PtjFt ¡ Ft¡n = x] = (1¡ w)º E[gtjFt ¡ Ft¡n = x]=

x

n(1¡ w)wº(1 + (1¡ wº) + (1¡ wº)2 + ¢ ¢ ¢) = 1

n(1¡ w)x

Suppose that a trading period corresponds to a month, the fundamental value has increased

by 50% over the last twelve months, and w = :75, as would be the case if 1=3 of a stock's

ownership was by type-d investors, each of them had a gain-related disposition e®ect that

in°uenced their demand function as much as the spread between the fundamental value and

the market price. Then, next month the price is expected to increase by 5048% or slightly over

1%.

Of course, despite our best attempts at arguing that this is similar to the size of the

Jegadeesh and Titman (1993) momentum e®ect, we have no way to assess if the 0:75 value

11If the econometrician can view the distribution of price changes between two consecutive dates within aninterval over which a price change has occurred as being symmetric with respect to any pair of consecutivedates, the above proposition goes through in exactly the same form with prices replacing fundamental values.Indeed, as long as expected price changes between two consecutive dates are of the same sign as the pricechange over the surrounding interval, positive past price changes generate positive expected returns in theperiod immediately following the interval and vice versa.

14

for w is truly reasonable. However, the model does provide other predictions, such as those

about volume in the next subsection.

F. Determinants of Equilibrium Trades and Trading Volume

To obtain an expression for trading volume in the model, substitute the equilibrium price

Pt, as given in equation (5), into the demand equations (1) and (2), and multiply the respec-

tive demands by 1¡¹ and ¹, respectively. This gives the equilibrium aggregate shareholdingof each investor-type as a function of the unrealized gain gt = Pt ¡Rt. Speci¯cally,

(1¡ ¹)Drt = 1¡ ¹+ ctgt; ¹Ddt = ¹¡ ctgtwhere ct = bt¸¹(1¡ ¹) is a positive parameter. It follows that the change in the aggregateequilibrium shareholdings of each investor-type is proportional to the change in the unrealized

gain. For type-r investors:

(1¡ ¹)(Drt+1 ¡Drt ) = ct(gt+1 ¡ gt) + (ct+1 ¡ ct)gt+1while for type-d investors:

¹(Ddt+1 ¡Ddt ) = ¡ct(gt+1 ¡ gt)¡ (ct+1 ¡ ct)gt+1 (22)

implying the following result:

Proposition 7 Assume that ct+1 ¡ ct is su±ciently small. Then type-d investors sell stockto type-r investors when their unrealized gain increases and buy stock from type-r investorswhen their gain decreases.

To elaborate on this point, assume ct = c and substitute equation (18) into the right

hand side of (22) to obtain

¹(Ddt+1 ¡Ddt ) = ¡cw(²t+1 ¡ ºtgt)

Hence, whether type-d investors buy or sell depends on the sign and the magnitude of the

innovation in the fundamental value, ²t+1, in relation to the hurdle ºtgt.

² Case 1: gt > 0. Then, at t+1, type-d investors sell (these \winners") only on su±cientlygood news. The bigger the gain gt, the better the news must be to induce a sale.

² Case 2: gt < 0. Then, at t+1, type-d investors buy (these \losers") only on su±cientlybad news. The more negative gt is, the worse the news must be to induce additional

purchase.

15

Trading volume, obtained by taking the absolute value of the prior equation is thus

Vt+1 = cwj²t+1 ¡ ºtgtj (23)

By substituting equation (9) into (23), we can also express the volume in terms of price

changes as opposed to changes in the fundamental value.

Vt+1 = c j(Pt+1 ¡ Pt)¡ ºtgtj = c¯̄̄̄(Pt+1 ¡ Pt)¡ Et[Pt+1 ¡ Pt]

1¡ w¯̄̄̄

This implies the following:

Proposition 8 Assume that ct+1 ¡ ct is su±ciently small. Then the volume associatedwith an increase in the market price is going to be larger when the reference price exceedsthe market price than when the market price exceeds the reference price. Also, the volumeassociated with an decrease in the market price is going to be larger when the market priceexceeds the reference price than when the reference price exceeds the market price.

Stocks that have generally been experiencing price increases are those for which the

market price tends to exceed the reference price and vice versa. Proposition 8 suggests that

major reversals of fortune are more likely to beget larger volume than repetitions of past

trends. That is, large price decreases (increases) will be associated with the greatest volume

impact for stocks that have experienced major and consistent increases (decreases) in value.

G. Numerical Findings and the Length of the Past Return Horizon for Mo-mentum

Two types of numerical simulations generated several interesting ¯ndings. These sim-

ulations assume bt is constant over time and that the reference price updating weight is

proportional to turnover.

In a binomial simulation of the model where each period F either goes up (\+" move)

or down (\¡" move) by a ¯xed amount:

² Fix n > m. Among paths with n \+" moves and m \¡" moves, the path with thehighest expected positive price change has all the \¡" moves at the beginning, whileEt[Pt+1 ¡ Pt] is small or even negative along paths for which all the \¡" moves occurat the end. One should avoid buying winners from the more distant past that recently

have begun to decline in value in a fairly persistent manner.

When the simulations are based on i.i.d. normal innovations in F :

16

² Stocks with high current volume and low past volume tend to have larger momentum.If Ft¡Ft¡n > 0, then Et[Pt+1¡Pt] is strongly positively correlated with Vt, and slightlynegatively correlated with average past volume over [t ¡ n; t]. If Ft ¡ Ft¡n < 0, thenEt[Pt+1¡Pt] is strongly negatively correlated with Vt and slightly positively correlatedwith average past volume over [t¡ n; t].

² Volume and absolute price change are strongly positively correlated.² The di®erence in 1-month returns of the top decile minus bottom decile of past per-

forming stocks, plotted as a function of past return horizon, has a humped shape. See

Figure 1.

This last ¯nding deserves special mention as one of the great curiosities of momentum

is that it only seems to function at intermediate horizons. Our model does not generate

reversals at short or long horizons, but it does generate less pro¯t from momentum when the

momentum portfolio is formed using past returns over short or long horizons. The model

suggests that the most pro¯table horizons generated from the model are those that use

intermediate horizon past returns for portfolio formation. The numerical ¯nding is di±cult

to prove analytically; however, it seems rather intuitive. Over very short horizons, it is

rather di±cult for the gain to deviate from zero by a large amount, as fundamentals have

not had much time to move, even among the best and worst performing stocks. While the

volatility of the change in a fundamental value that follows a random walk is proportional

to the square root of the past horizon's length, one must also consider how horizon a®ects

the stochastic process for the reference price. The reference price reverts to the fundamental

value. Over short horizons, such reversion cannot have much of an e®ect. However, over

a long horizon, reversion to the fundamental value is likely to have a tremendous e®ect.

Indeed, as we learned earlier, the gain will be zero with probability one. Reversion in the

gain to zero only enhances the frequency with which this occurs.

A good analogy is a race between two thoroughbred horses with equal expected speed:

the fundamental value horse and the reference price horse. When we sort on past winners,

we are saying that the fundamental value horse is in the lead. However, both shortly after

the start of the race, and towards the end of the race, he cannot have a very big lead. Near

the start, (and even assuming that the initial acceleration from the starting gate took place

instantly), the top decile fundamental value horse has not had enough time to get far ahead,

despite the average speed being greatest for this leg of the race. Moreover, throughout the

race, whenever the fundamental value horse gets far ahead, the reference price horse speeds

up. However, the fundamental value horse's speed is not persistent. Hence, conditional on

him being far ahead in the ¯rst part of the race, he is likely to slow down. This means that

the point at which the gap between the top decile fundamental value horse and the reference

price horse is likely to be greatest is somewhere in the middle of the race.

17

II. Empirical Tests

Our empirical work utilizes weekly returns, turnover (weekly trading volume divided

by the number of outstanding shares), and market capitalization data from the MiniCRSP

database. The dataset includes all ordinary common shares traded on the NYSE and AMEX

exchanges. NASDAQ ¯rms are excluded because of multiple counting of dealer trades. The

sample period, from July 1962 to December 1996, consists of 1799 weeks.

A. Regression Description

We analyze the average slope coe±cients of weekly cross-sectional regressions and their

time series t-statistics, as in Fama and MacBeth (1973). The week t return of stock j,

rjt =P jt ¡P jt¡1P jt¡1

, is the dependent variable. Denote rjt¡t2:t¡t1 as stock j's cumulative return from

weeks t¡t2 to t¡t1. The prior cumulative returns over short, intermediate, and long horizonsare used as control regressors for the return e®ects described in Jegadeesh (1990), Jegadeesh

and Titman (1993), and DeBondt and Thaler (1995). Regressor sjt¡1, the logarithm of ¯rm

j's market capitalization at the end of week t¡ 1, controls for the return premium e®ect of

¯rm size. We also control for the possible e®ects of volume, including those described in Lee

and Swaminathan (2000) and Gervais, Kaniel, and Minelgrin (2001), by including ¹V jt¡52:t¡1,stock j's average weekly turnover over the 52 weeks prior to week t as a regressor (and in

later regressions, three interaction terms, computed as the product of the former volume

variable and returns over the three past return horizons). We then study the coe±cient on

gjt¡1, a capital gains related proxy. Formally, we analyze the regression,

r = a0 + a1r¡4:¡1 + a2r¡52:¡5 + a3r¡156:¡53 + a4 ¹V + a5s+ a6g (24)

and variants of it, where, for brevity, we have dropped j superscripts and t subscripts.

Recall that the theoretical model states that

Et¡1

·Pt ¡ Pt¡1Pt¡1

¸= (1¡ w)ºt¡1 Pt¡1 ¡Rt¡1

Pt¡1

This equation suggests that a measurable variable that predicts expected returns is the

percentage di®erence between the market price and the reference price at the beginning of

week t. Our proxy for this variable, the capital gains overhang, is

gt¡1 =Pt¡2 ¡Rt¡1

Pt¡2

Theory says that this key regressor should employ Pt¡1 instead of Pt¡2. We lag the mar-ket price by one week to avoid confounding market microstructure e®ects, such as bid-ask

bounce.12

12We obtain essentially the same results when we multiply the gain variable by turnover, as speci¯ed

18

B. Specifying a Reference Price

Theory allows the reference price to be any weighted average of historical market prices.

In empirical work, we have to specify how the weights used to update the reference prices

are determined. Because our theory was motivated by the disposition e®ect, we believe

that reference prices should represent the best estimate of a stock's cost basis to disposition

investors. Since we cannot identify who these disposition investors are, our proxy for the

cost basis of a stock is an estimate of the aggregate cost basis for all outstanding shares.

The date t aggregate basis thus requires us to use price and volume data to estimate the

fraction of shares purchased at date t ¡ n < t, at price Pt¡n, that are still held by theiroriginal purchasers at date t. Summing the products of these fractions and the prices at the

relevant prior dates generates the aggregate cost basis for the market.13

Estimating these fractions requires us to model trading behavior. We model the fraction

of shares purchased at week t¡ n < t and held by the week t¡ n purchaser through week tas given by

Vt¡nn¡1Y¿=1

[1¡ Vt¡n+¿ ]

If we truncate the reference price estimation process at the price ¯ve years prior to week t,

the date t reference price

Rt =1

k

260Xn=1

ÃVt¡n

n¡1Y¿=1

[1¡ Vt¡n+¿ ]!Pt¡n (25)

where the scaling constant that makes the fractions sum to one,

k =260Xn=1

Vt¡n

Ãn¡1Y¿=1

[1¡ Vt¡n+¿ ]!

This model is equivalent to assuming that all shares are symmetric. That is, irrespective of

its trading history, each outstanding share is equally likely to be sold at any date. It can be

shown that with a constant weekly turnover of V , the average holding period is 1Vweeks.

above. We opt for the more parsimonious representation, which omits this factor, because there may be across-sectional relation between a ¯rm's typical ºt and w, which we cannot estimate.13To gain a sense of how good the aggregate cost basis is a proxy for the reference price of the disposition

agents, we do the following theoretical experiment. First, simulate 10,000 path of economy for 260 weeksaccording to our theoretical model, assuming that the stock is fairly valued at the initial date, and that bt isa constant derived from the excess demand function of an investor with myopic exponential utility function.Then, along each path, compute the cost basis of the disposition agent as well as the aggregate cost basis ofboth type of investors as given by (25), and calculate their correlation. We tried a wide range of parameterschoices (¹; ¸; ¾ and risk aversion °), and ¯nd that the two cost basis are always highly correlated, with meanand median correlation above 0.9 in all cases.

19

Assuming a constant weekly turnover of 1%, which is approximately the mean for entire

sample period, this implies an average holding period of 2 years.

As noted earlier in the paper, the logic behind the expression for the reference price in

equation (25) is straightforward if we assume k = 1, as is the case when the sum is in¯nite

rather than over 260 weeks. The turnover ratios, the V s, are then probabilities and each of

the bracketed factors inside the product symbol represents the probability that a share did

not trade at date t ¡ n + ¿ ; the term in front of the product symbol, Vt¡n, represents theprobability that the share traded at date t ¡ n; the term in large parentheses in equation

(25) is the probability that the share's basis is the price at date t ¡ n; and the sum is the

expected cost basis.

Observe that more recent trading prices have more weight on the reference price, other

things equal. This is because the survival probability for a historical price declines geometri-

cally with the passage of time. Indeed, distant prices negligibly in°uence the reference price.

Recognizing that distant market prices have little in°uence on the regressor, we truncate

the estimation at ¯ve years and e®ectively rescale the weights to sum to one by having a

k < 1. This allows us to estimate the reference price in a consistent manner across the

sample period. The 5-year cuto®, while arbitrary, allows us to analyze a reasonable portion

of our sample period: July 1967 on. Stocks that lack at least ¯ve years of historical return

and turnover data at a particular week are excluded from the cross-sectional regression for

that week. 14

C. Summary Statistics

Figure 2 plots the weekly time series of the 10th, 50th, and 90th percentile of the capital

gains regressor. It indicates that there is wide cross-sectional dispersion in this regressor and

a fair amount of time series variation as well. For most ¯rms, the time series of this variable

exhibits signi¯cant comovements with the past returns of the S&P 500. For the 10th, 50th,

and 90th percentile of the regressor, plotted in Figure 1, the correlations between the weekly

time series of the regressor and the past one-year percentage change in the S&P 500 index

are respectively 0.50, 0.60, and 0.62.

Table 1 Panel A reports summary statistics on each of the variables used in the regression

described above. These include time series means and standard deviations of the cross-

sectional averages of the dependent and independent variables, along with time series means

of their 10th, 50th and 90th percentiles.

We obtain further insight into what determines the critical capital gains regressor by

regressing it (cross-sectionally) on stock j's cumulative return and average weekly turnover

for three past periods: very short term (de¯ned as the last four weeks), intermediate horizon

14We veri¯ed that our regression results remain about the same when return and turnover data over threeor seven prior years are used to calculate the reference price.

20

(between one month and one year ago) and long horizon (between one and three years ago).

Size is also included as a control regressor. Panel B of Table I reports that, on average,

about 59% of the cross-sectional variation in the capital gains variable can be explained

by di®erences in past returns, past turnover, and ¯rm size. Earlier, we explained that the

reference price is always trying to catch up to a fundamental value that deviates from the

reference price for large return realizations. Consistent with this, Panel B shows that our

capital gains variable, in both cases, is positively related to past returns and negatively

related to past turnover.15 Also, consistent with the thoroughbred horse analogy explaining

why intermediate horizons are most important, we ¯nd that the e®ect of intermediate horizon

turnover on the capital gains variable is much stronger than the e®ect of turnover from the

other two horizons. Controlling for past returns, a low volume winner has a larger capital

gain, while a high volume loser has a larger capital loss. Finally, the size coe±cient in this

regression is signi¯cantly positive, perhaps re°ecting that large ¯rms have grown in the past

at horizons not captured by our past return variables and thus tend to have experienced

larger capital gains.

D. Expected Returns, Past Returns, and the Capital Gains Overhang

Table 2 presents the average coe±cients and time-series t-statistics for the regression

described by equation (24) and variations of it that omit certain regressors. Each panel

reports average coe±cients and test statistics for all months in the sample, for January

only, for February-November only, and for December only. All panels include the ¯rm size

regressor. Panel A employs only the three past return regressors. Panel B adds volume as a

fourth regressor. Panel C adds the capital gains overhang as a ¯fth regressor.

Panels A and B contain no surprises. As can be seen, when the capital gains overhang

variable is excluded from the regression, there is a reversal of returns at both the very short

and long horizons, but continuations in returns over the intermediate horizon. Consistent

with prior research, the long horizon reversal appears to be due to January. Panel B indicates

that there is a volume e®ect, albeit one that is hard to interpret, but it does not seem to

alter the conclusion about the horizons for pro¯table momentum and contrarian strategies.

Panel C is rather astounding, however. When the capital gains overhang regressor is

included in the regression, there is no longer an intermediate horizon momentum e®ect. The

coe±cient, a2, is insigni¯cant , both overall and from February through November. However,

except for January, there is a remarkably strong cross-sectional relation between the capital

gains overhang variable and future returns, with a sign predicted by the model.

15The mean, median and standard deviation of the time series of correlations between ¯rm's capital gainoverhang and past one year return in the cross-section over our smaple period is 0.5482, 0.5529 and 0.1250respectively.

21

E. Explaining Seasonalities

The seasonalities observed in Table II are consistent with what other researchers have

found.16 They are fairly easy to explain within the context of our theoretical model if we

accept that there is an additional perturbation in demand arising from tax loss selling.

Grinblatt and Keloharju (2001), for example, found that there was no disposition e®ect

in December, and attributed this to the marginal impact of tax loss selling. If we generalize

the demand function of the disposition investor,

Ddt = 1 + bt[(Ft ¡ Pt) + ¸t(Rt ¡ Pt)] (26)

and assume that ¸t drifts downward in December for certain, as might be expected because of

tax loss selling (possibly, but not necessarily, becoming negative) and reverts to its normal

positive value sometime in early January, we would ¯nd that the equilibrium e®ects of

this seasonal demand perturbation would be consistent with our empirical ¯ndings. The

downward drift in ¸ in December implies that market prices move closer to fundamental

values. For stocks with capital losses, implying that the fundamental value is below the

market price, convergence towards the fundamental value from the decline in ¸ represents

an added force that makes the market price decline even further than it would were ¸ to

remain constant. Similarly, the increase in ¸ in early January would make the prices of these

same stocks with capital losses deviate again from their fair values, leading to a January

reversal.

To understand this more formally, note that with the generalized disposition demand,

equation (26), the expected price change, formerly in equation (14), generalizes to

Et[Pt+1 ¡ Pt] =µ(1¡ wt)ºt + (wt+1 ¡ wt)(1¡ ºtwt)

wt

¶(Pt ¡Rt)

Hence, if we know that ¸t+1 is going to be lower than ¸t, which makes wt+1¡wt positive, theexpected return between dates t and t + 1 is going to be larger. The evidence in Grinblatt

and Keloharju (2001) suggests that over the course of December, ¸ declines to zero but is

positive during the rest of the year. Viewed from the end of November, this would be like

knowing that wt+1 = 1 and larger than wt, thus generating a larger coe±cient on the gain

regressor in December than would be observed in months with wt+1 = wt. Viewed from the

end of December, wt = 1 and larger than wt+1. This makes the expected price change during

January negatively related to the gain regressor.

16For example, momentum strategies that form portfolios from past returns over intermediate horizonsappear to be most e®ective in December, and there is a strong January reversal in the direction of expectedreturns when using past returns over any horizon. See, for example, Jegadeesh and Titman (1993), Grundyand Martin (2001) and Grinblatt and Moskowitz (2001).

22

F. Robustness Across Subperiods and Gain De¯nitions

To most observers, the ¯rst and second half of our sample period present di®erent por-

traits of the stock market. From July 1967 to March 1982, average returns were low, liquidity

was low, and trading costs including commissions were high. The second half of our sam-

ple period, April 1982 to December 1996 corresponds to a sea change in the stock market.

Beginning in August 1982, average returns and trading volume appeared to explode and

trading costs rapidly declined. These subperiods also demarcate an important turning point

in the strength of the ¯rm size e®ect. In the second half of our sample period, size was far

less important as a determinant of return premia. Despite these di®erences, if our theory

is part of the core foundation of equilibrium pricing, there should be little di®erence in the

coe±cient on our capital gains regressor. Panels D and E of Table II con¯rm this hypothesis.

There is only about a one standard error di®erence between the average coe±cients on the

capital gains regressor in the two subperiods. In both subperiods, the average coe±cient is

highly signi¯cant.

Although we do not report this formally in a table, the signs and signi¯cance of the capital

gains overhang regressor are not drastically altered by restricting the sample to various size

quintiles, either. Alternative de¯nitions of the capital gain percentage, such as

gt¡1 =Pt¡2 ¡Rt¡1

Rt¡1or

Pt¡2 ¡Rt¡2Pt¡2

are also signi¯cantly related to the future return and knock out past returns over intermediate

horizons as a signi¯cant predictor of future returns.

G. Alternative Explanations

Could the strength of the capital gains variable as a predictor of returns be due to some

alternative explanation? Table III investigates this issue with respect to two alternatives.

First, Panels A and B examine whether there is some sort of interaction between a ¯rm's

average historical turnover and future returns. For example, the results in Table II Panel

C may have arisen because cross-sectional di®erences in liquidity imply that the reference

prices of more liquid stocks place greater weight on more recent prices than the reference

prices of less liquid stocks. By formulating a reference price using the average turnover over

the past year in place of each week's actual turnover, we assess whether it is only the cross-

sectional di®erence in liquidity that is responsible for the predictive power of our original

gain variable, or whether the information about a stock's capital gain inherent in the time

series of its historical weekly turnover also contributes to the predictive power of our ¯ndings

in Table II.

In Panels A and B of Table III, we compute an alternative week t reference price using¹V jt , ¯rm j's average weekly turnover from weeks t ¡ 52 to t ¡ 1 for all of the 260 V s in

23

equation (25). Panel A replicates Panel C of Table II, except that in place of the gain

variable, we compute an alternative gain variable using the alternative reference price. As

Panel A indicates, using a ¯rm's average turnover for the reference price computation instead

of the actual weekly turnover generates a signi¯cant coe±cient on the gain variable. The

results are similar to those of Table II Panel C, in that past returns have no predictive power.

Moreover, the coe±cients and t-statistics on the alternative gain variable are similar to those

in Table II Panel C.

Table III Panel B runs a horse race between the two gain variables. It is identical to Table

III Panel A, except that the Table II proxy for ¯rm j's capital gain is added as a regressor.

The inclusion of this variable eliminates the signi¯cance of the alternative gain variable, and

its coe±cient is about the same size as that in Table II Panel C. While our original gain

variable is based on an imperfect model of the actual capital gains overhang in the market,

it is probably a more precise estimate of aggregate capital gains than the alternative capital

gains proxy constructed from average historical turnover. The fact that it \knocks out" the

alternative as a predictor of future returns is consistent with more precise estimates of the

aggregate capital gain being better predictors of future returns.

A second concern about the signi¯cance of our capital gains proxy in Table II is that it

represents some complicated interaction between volume and past returns. For example, Lee

and Swaminathan (2000) suggested that high volume losers should have lower returns than

average volume losers and empirically documented that this was indeed the case. Hence, it

is possible that our ¯ndings in Table II arise from the capital gains overhang variable being

correlated with some interaction between intermediate horizon past returns and volume.

Panels C and D of Table III test this hypothesis by adding three turnover and past return

interaction terms.

Table 3 Panel C analyzes the impact of these regressors in the absence of a capital gains

regressor. Even though two of the three turnover-return coe±cients are signi¯cant, the

inclusion of these regressors does not subsume the intermediate horizon momentum e®ect.

Rather, the volume-return interaction seems to work in part by moderating the strong one-

month return reversal.

Once the capital gains variable is added to the regression, as in Table III Panel D, the

intermediate horizon past return becomes insigni¯cant, while the capital gains coe±cient

is highly signi¯cant. Comparing Table II Panel C with Table III Panel D, the average

regression coe±cient for the capital gains variable and its t-statistic are almost unchanged

in the presence of the three turnover and past return interaction terms.

24

III. Relation to Prior Research

Jegadeesh and Titman (1993) popularized the notion that strategies of buying stocks

with high returns over the prior three to twelve months and selling stocks with poor returns

over the same past horizon dominates a buy and hold strategy,17 Ever since then, researchers

have attempted to come up with explanations for the phenomenon.

Conrad and Kaul (1998) argue that the pro¯tability of momentum strategies could be due

to cross-sectional variation in the unconditional expected returns rather than any predictable

time-series variation in stock returns. Yet Jegadeesh and Titman (2000) ¯nd that the cu-

mulative return in months 13 to 60 after the formation of momentum portfolio is negative,

which is inconsistent with the Conrad and Kaul hypothesis. Moskowitz and Grinblatt (1999)

show the component of momentum pro¯ts due to cross-sectional variation in unconditional

expected returns is small. Grundy and Martin's (2001) evidence also appears to contradict

this hypothesis. They ¯nd that the risk-adjusted pro¯tability of a total return momentum

strategy is more than 1:3% per month and remarkably large and stable across subperiods,

even after subtracting each stock's mean return from its return during the investment period.

Moskowitz and Grinblatt (1999) ¯nd that a signi¯cant component of momentum can be

explained by industry e®ects. However, this does not mean that individual stock momentum

does not exist. Moskowitz and Grinblatt (1999), Grundy and Martin (2001), and Chordia

and Shivakumar (2000) show that a component of individual stock momentum is distinct

from industry momentum. The latter paper also argues that momentum pro¯ts are driven

by time varying conditional expected returns that are related to the business cycle.

Another strand of the literature uses behavioral models to explain momentum pro¯ts.18

These models can be divided into two camps, depending on whether investor behavior gen-

erates overreaction or underreaction. In the positive feedback trader model of DeLong et al

(1990b), prices initially overreact to news about fundamentals, and continue to overreact for

a period of time. Daniel, Hirshleifer and Subrahmanyam (1998) present a model where in-

vestors are overcon¯dent. This implies overeaction to private information and underreaction

to public information arrival. The investors also su®er from a self-attribution bias. Their

17This ¯nding appears to be fairly universal and robust to methodological tweaking. Rouwenhorst (1997),for example, ¯nds that momentum strategies work in twelve European markets. Chui, Titman, and Wei(2000) document that with the exception of Japan and Korea, momentum pro¯ts can be earned in Asianmarkets. Jegadeesh and Titman (2000) document that momentum pro¯ts persisted throughout the 1990s.In contrast, other well known anomalies such as small ¯rm e®ect and book-to-market e®ect disappeared afterbeing well-publicized. Jegadeesh and Titman (1993) and Fama and French (1996) ¯nd that risk adjustmenttends to accentuate momentum pro¯ts. Chan, Jegadeesh and Lakonishok (1996) show that intermediatehorizon return continuation can be partially explained by underreaction to earnings news but that pricemomentum is not subsumed by earnings momentum. Lee and Swaminathan (2000) show that past tradingvolume predicts both the magnitude and the persistence of future price momentum.18Hirshleifer (2001) gives a comprehensive account of psychological biases and empirical evidence on the

importance of investor psychology for security prices.

25

behavior generates delayed overreaction to the information which is eventually reversed.

Barberis, Shleifer and Vishny (1998) argue that the representative heuristic19 may lead

investors to extrapolate current earnings growth well into the future. At the same time,

investors' conservativism bias leads to underreaction to new public information. In Hong

and Stein (2000), agents can use only part of the information about the economy because

of communication frictions. In their model, private information di®uses slowly through the

population of investors, which causes underreaction in the short run. Momentum traders

can pro¯t by trend-chasing, but cause overreaction at long horizons in doing so.

Our explanation of the pro¯tability of momentum strategies is distinct from explanations

in prior research. Our investors have no cognitive biases such as those based on overcon-

¯dence, self attribution, conservativism, or representativeness. There is no mistaken belief

about the fundamental value. There is no time variation in risk, risk aversion, or investor

sentiment driving our results. There are no hidden factors such as those based on indus-

try. Information is symmetric. In asymmetric information models, trading volume re°ects

investors' disagreements about a stock's intrinsic value, and often requires the existence of

noise traders to generate trading volume. There is no information asymmetry in our model,

and no noise traders, but there is volume. Trading occurs because some investors are sub-

ject to the disposition e®ect. Past trading volume a®ects the equilibrium price through its

in°uence on the reference price, while most extant models have a representative agent and

volume plays no role.

Most importantly, our model is based on well-documented investor behavior and princi-

ples of psychology. Disposition behavior, while inconsistent with the standard neoclassical

framework, has been justi¯ed as a consequence of theories of behavior including prospect

theory,20 regret theory,21 and cognitive dissonance theory.22 Camerer and Weber (1998) and

Heilmann, Lager and Oehler (2000) found evidence for disposition behavior in experimental

markets. Evidence of disposition behavior among actual investors is found in Odean (1998),

Heath, Huddart and Lang (1999), Grinblatt and Keloharju (2001), and Locke and Mann

(1999). Odean (1998) analyzes accounts at a large brokerage house and found that there

was a greater tendency to sell stocks with paper capital gains than those with paper losses.23

Grinblatt and Keloharju (2001) ¯nd a similar e®ect among all types of investors in Finland,

even after controlling for a variety of variables that may determine trading. They also ob-

serve that the disposition behavior interacts with past returns in a multiplicative fashion

and has a pronounced seasonality: it disappears in December. Using data from a major

Israeli brokerage house during 1994, Shapira and Venezia show that both professional and

19See Tversky and Kahneman (1974).20See Shefrin and Statman's (1985) interpretation of Kahneman and Tversky (1979).21See Shefrin and Statman (1985)22See Shefrin and Statman (1985), Camerer and Weber (1998).23Using the same dataset, Ranguelova (2001) reports that the disposition e®ect is concentrated primarily

in large cap stocks and the relationship between ¯rm size and the disposition e®ect appear to be monotonic.

26

independent investors exhibit the disposition e®ect, although the e®ect is stronger for inde-

pendent investors. Heath, Huddart and Lang (1999) uncover disposition behavior relative to

a reference price of a prior high for the stock price by studying the option exercise behavior

of over 50,000 employees at seven corporations. Locke and Mann (1999) present evidence for

the existence of a disposition e®ect within a sample of professional futures traders. In their

study, traders held losing trades longer than winning trades and average position sizes for

losing trades were larger than for winners. Ferris, Haugen and Makhija (1988) argue that a

disposition e®ect has to exist by studying the relationship between volume at a given point

in time with historical volume at di®erential prices, controlling for seasonal e®ects to isolate

tax motivated trading. The disposition e®ect also in°uences agents in the IPO and housing

markets.24

There are a set of papers that are linked to modeling how loss aversion a®ects equilibrium

prices and portfolio holdings.25 Barberis, Huang and Santos (2001) model investor prefer-

ences to re°ect a combination of loss aversion and the \house money" e®ect of Thaler and

Johnson (1990). Their goal is to show how changing risk aversion explains the high mean,

high volatility, and signi¯cant predictability of stock returns. They have a representative

agent and no trading, but ¯nd that the variation in risk aversion of their representative agent

allows returns to be much more volatile than the underlying dividends. Moreover, asset re-

turn predictability found in their model is consistent with the pro¯tability of contrarian

strategies. Our model, by contrast, is consistent with contrarian strategies being money

losers and our investors limited willingness to take positions is consistent with risk aversion.

Trading arises only because of the disposition e®ect and plays an important role in generating

momentum. Although both the house money e®ect and the disposition e®ect can be rooted

in prospect theory, the fact that they lead to such opposite results suggests that they are

truly distinct phenomena. Barberis and Huang (2001) extend this paper to multiple assets in

order to address cross-sectional expected return patterns, such as the value premium. Ang,

Berkaert, and Liu (2001) study portfolio choice under the disappointment aversion prefer-

ence, where outcomes below the certainty equivalent are weighted more heavily than above

the certainty equivalent. They study optimal \non-participation" in the stock market and

cross-sectional variation in portfolio holdings, and they contrast their preference structure

with loss aversion, which, for many parameter values leads to troublesome portfolio predic-

tions. Both Gomes (2000) and Berkelaar and Kouwenberg (2000) study optimal portfolio

choice under loss aversion. Both papers ¯nd that demand under loss aversion shares some

common features with the disposition e®ect, pointing to the possibility that loss aversion

can be consistent with the disposition e®ect.

24See Case and Shiller (1988), Genesove and Mayer (2001) and Kaustia (2001).25In empirical work, Coval and Shumway (2001) document Chicago Board of Trade proprietary futures

traders are highly loss averse, as they assume signi¯cantly more afternoon risk following morning losses thanfollowing morning gains.

27

IV. Conclusion

Our paper has developed a model of equilibrium asset prices based on the disposition

e®ect. By restricting loss realization aversion to be a geometric deviation from fully rational

behavior, we are able to generate closed form solutions and a set of powerful propositions

about conditional expected future returns without solving the optimal dynamic portfolio

problem of rational agents. The solution to the portfolio problem drops out of the equations

we are interested in, which focus on deviations from the rational norm. We then test the

model and surprisingly show that the Jegadeesh and Titman (1993) momentum e®ect largely

disappears. This suggests that it is the correlation between past returns and variables related

to the disposition e®ect that may be driving momentum in stock returns.

In the model presented here, the critical variable determining the sign and magnitude of

a stock's expected return is the di®erence between the stock's market price and its reference

price. Despite having fully rational investors in the model, they cannot eliminate the impact

of the gain on equilibrium prices. Although, on average, the gain shrinks, the payo® to more

rational investors is uncertain. Hence, rational investors cannot ascertain when reference

prices, and hence market prices, will converge to fundamental values.

DeLong et al (1990b) show that when there are positive feedback traders in the economy,

rational arbitrageurs who anticipate their impact on demand can front-run the other investors

and destabilize prices. Speci¯cally, when the rational investor receives good news today,

he buys more shares to drive up the price. This, in turn, attracts the positive feedback

investors who buy tomorrow so that the rational investor can exit with a pro¯t. In our

model, there is no way to anticipate the disposition demand in advance, as it is determined

by the future realization of the fundamental value, which follows a random walk and hence

is unpredictable. Any degree of risk aversion on the part of rational agents thus prevents

the model from collapsing.

Our model falls in the class of \underreaction models." However, it also points out some

interesting implications of underreaction and suggests that our ¯eld may have to better

clarify what we mean by the term. For example, Barberis et al. (1998) and Shleifer (2000)

de¯ne underreaction as occurring when the average return on the stock following good news

is higher than the average return following bad news. Our model clearly has underreaction,

but the ¯t with this de¯nition is imperfect because path dependency generates cases where

this de¯nition does not hold.26 Our model points to the di±culty of measuring underreaction

26The future expected return in our model is of the same sign as the current gain. If the gain is negativenow, it will continue to be negative even after good news is announced, assuming that the news is not goodenough. Hence the expected return may be negative after the announcement of good news following pathswith capital losses. On the other hand, if the current gain is positive, it may still be positive and hence theexpected return is positive, even if the news is bad (but not too bad). Hence, because the path associatedwith the good news had a capital loss and the path associated with the bad news had a capital gain, theexpected return after good news was lower than the expected return following bad news. While this is

28

(or overreaction) in terms of subsequent price changes without being able to measure the

degree of underreaction (or overreaction) existing in the market at all times used for these

computations. Similarly, special cases of our model have underreaction at all times, but no

tendency for prices to converge to fundamental values. In these cases, there is not even the

positive autocorrelation that is typically associated with underreaction.

Our assumptions are quite general, allowing the model's analysis of a partial equilib-

rium for a single asset to be consistent with more comprehensive modeling of a multi-asset

equilibrium. The process by which the market arrives at a fair value in an intertemporal

multi-asset economy can be quite complicated, but that is not our concern. We simply want

to understand as clearly and analytically as possible how a perturbation of investors' demand

for a single stock, due to the disposition e®ect, generates deviations from the fully rational

equilibrium. Our \partial" equilibrium approach should not generate conclusions that di®er

from those obtained by postulating utility functions and solving for demand functions over

multiple assets that optimize utility unless momentum strategies are true arbitrages that lack

risk. However, prior empirical evidence indicates that momentum strategies are quite risky.

Moreover, we have found that the capital gains overhang of individual stocks is strongly

correlated with the cumulative past return of a broad market index like S&P 500. Thus the

disposition components of demand across stocks are also likely to be positively correlated,

making the disposition e®ect a systematic risk to potential arbitrageurs. In short, we believe

that our approach, despite lacking a closed form solution for demand functions, does not

generate aberrational conclusions.

Similarly, using what e®ectively are two representative agents is an oversimpli¯cation.

However, such a simpli¯cation is reasonable if aggregate demand generates e®ective aggregate

reference prices that are weighted averages of current prices and past aggregate reference

prices. While this aggregation cannot be done analytically, we have been quite general in

allowing weights for reference prices to time vary and be path dependent. With this level

of generality in reference price construction, we would be surprised if the two representative

agent paradigm used here does not hold up to closer scrutiny.

The generality of the reference price updating rule in the paper's theoretical section is

both a strength and a weakness. Any attempt at de¯ning an aggregate reference price for

empirical work requires a concrete updating weight, with little guidance from our theory.

Real-world equilibria, with multiple investors, requires aggregation of investors' demand and

reference prices. Our speci¯cation of the aggregate reference price for empirical work is

not the only solution to this problem. While we have analyzed modest variations in the

reference price updating rule and found nothing to refute our conclusions, the exploration of

appropriate and alternative reference price rules is certainly an interesting avenue for future

research.

generally not the case, it points to the need for more precision in any de¯nition of underreaction.

29

Table ISummary Statistics

This table presents summary statistics of weekly data on NYSE and AMEX securities from July 1967 toDecember 1996, obtained from mini-CRSP. Panel A provides time series averages of the cross-sectionalmean, median, standard deviation, and 10th, 50th, and 90th percentiles of each of the variables used in theregression

r = a0 + a1r¡4:¡1 + a2r¡52:¡5 + a3r¡156:¡53 + a4 ¹V ++a5s+ a6g

where r is the week t return, r¡t1:¡t2 is the cumulative return from week t ¡ t1 through t ¡ t2; ¹V is theaverage weekly turnover ratio over the prior 52 weeks, the ratio of the week's share volume to the numberof outstanding shares; s is log(market capitalization) measured at the beginning of week t; g is the capitalgains regressor, computed as one less the ratio of the beginning of week t ¡ 1 reference price to the end ofweek t¡ 2 price, where the week t¡ 1 reference price is the average cost basis calculated from the formula

Rt¡1 =1

k

260Xn=1

ÃVt¡1¡n

n¡1Y¿=1

[1¡ Vt¡1¡n+¿ ]!Pt¡1¡n

with k a constant that makes the weights on past prices sum to one. Panel B presents more detaileddata on the association between the capital gains regressor and other variables. It contains the time-seriesaverage of the coe±cients and their associated time series t-statistics for 1539 weekly Fama-MacBeth typecross-sectional regressions of the form

g = a0 + a1r¡4:¡1 + a2r¡52:¡5 + a3r¡156:¡53 + a4V¡4:¡1 + a5V¡52:¡5 + a6V¡156:¡53 + a7s

where V¡t1:¡t2 is the average weekly turnover from t¡ t1 through t¡ t2. R2adj is the average of the weeklycross-sectional regression R2s adjusted for degrees of freedom.

Panel A: Time series average of summary statics of the regressors in the regression

r = a0 + a1r¡4:¡1 + a2r¡52:¡5 + a3r¡156:¡53 + a4 ¹V + a5s+ a6g

r¡4:¡1 r¡52:¡5 r¡156:¡53 ¹V s gMean 0.0119 0.1493 0.3487 0.0092 18.7207 0.0560Median 0.0045 0.0940 0.2098 0.0072 18.7251 0.1062Std 0.1073 0.4192 0.7585 0.0079 1.9441 0.2508

10 percentile -0.0959 -0.2538 -0.3227 0.0025 16.1399 -0.281090 percentile 0.1223 0.5816 1.1097 0.0181 21.2322 0.3122

Panel B: Average coe±cients and t-statistics (in parentheses) for the regressiong = a0 + a1r¡4:¡1 + a2r¡52:¡5 + a3r¡156:¡53 + a4V¡4:¡1 + a5V¡52:¡5 + a6V¡156:¡53 + a7s

a1 a2 a3 a4 a5 a6 a7 R2adj0.5527 0.4907 0.1771 -0.9159 -6.4051 -2.7843 0.0504 0.5879(73.0290) (51.7965) (37.5209) (-7.6351) (-45.0322) (-27.8215) (55.9642)

30

Table IICross-sectional Regression Estimates

This table presents the results of Fama-MacBeth (1973) cross-sectional regressions run each week on NYSEand Amex securities from July 1967 to December 1996. The weekly cross-sectional regressions include allstocks that have at least ¯ve years of historical trading data on mini-CRSP. The cross section of stock returnsin week t, denoted r, are regressed on a constant and some or all of the following variables: r¡t1:¡t2 = thecumulative return from week t¡t1 through t¡t2, computed over three past return horizons; ¹V = the averageweekly turnover ratio over the prior 52 weeks, with turnover being the ratio of the week's share volume tothe number of outstanding shares; s = log(market capitalization) measured at the beginning of week t; andg = the capital gains regressor, computed as one less the ratio of the beginning of week t¡ 1 reference priceto the end of week t¡ 2 price, where the week t¡ 1 reference price is the average cost basis calculated fromthe formula

Rt¡1 =1

k

260Xn=1

ÃVt¡1¡n

n¡1Y¿=1

[1¡ Vt¡1¡n+¿ ]!Pt¡1¡n

with k a constant that makes the weights on past prices sum to one. There are a total of 1539 weeklyregressions. The parameter estimates and t-statistics (in parentheses) are obtained from the time seriesof the corresponding cross-sectional regression coe±cients. We report the results of regressions over allmonths, for January only, February through November only, and December only. Panel A omits the capitalgains and turnover variables. Panel B omits the capital gains variable. Panel C contains the full set ofregressors. Panels D and E report results for the full set of regressors over the ¯rst and second half of thesample period.

Panel Art = a0 + a1r¡4:¡1 + a2r¡52:¡5 + a3r¡156:¡53 + a4s

Period a1 a2 a3 a4All -0.0482 0.0012 -0.0005 -0.0004

(-35.6415) (2.9527) (-3.0054) (-4.2733)Jan -0.0700 -0.0087 -0.0068 -0.0040

(-9.6647) (-4.5972) (-6.6744) (-10.9146)Feb-Nov -0.0459 0.0018 -0.0001 -0.0001

(-34.0613) (4.3344) (-0.6243) (-1.4488)Dec -0.0491 0.0051 0.0015 0.0008

(-9.9440) (3.8921) (2.8930) (3.0164)

Panel Br = a0 + a1r¡4:¡1 + a2r¡52:¡5 + a3r¡156:¡53 + a4 ¹V + a5s

Period a1 a2 a3 a4 a5All -0.0488 0.0014 -0.0005 -0.0540 -0.0004

(-37.2470) (3.5703) (-2.6700) (-2.5732) (-4.4200)Jan -0.0706 -0.0086 -0.0069 0.0681 -0.0042

(-9.7366) (-4.5561) (-6.5561) (0.9793) (-11.2309)Feb-Nov -0.0465 0.0021 -0.0000 -0.0729 -0.0001

(-36.0594) (5.1324) (-0.1979) (-3.1591) (-1.5202)Dec -0.0489 0.0049 0.0015 0.0088 0.0009

(-10.2429) (3.7745) (2.8046) (0.1214) (3.1917)

31

Panel Cr = a0 + a1r¡4:¡1 + a2r¡52:¡5 + a3r¡156:¡53 + a4 ¹V + a5s+ a6g

Period a1 a2 a3 a4 a5 a6All -0.0425 -0.0002 -0.0007 -0.0188 -0.0004 0.0040

(-35.9364) (-0.6794) (-5.0871) (-0.9364) (-5.2885) (7.7885)Jan -0.0520 -0.0001 -0.0025 -0.0620 -0.0026 -0.0117

(-10.9905) (-0.0477) (-3.8964) (-0.9768) (-8.4381) (-4.9519)Feb-Nov -0.0407 -0.0000 -0.0006 -0.0291 -0.0002 0.0050

(-32.6251) (-0.0768) (-3.6950) (-1.3143) (-2.8816) (9.4191)Dec -0.0498 -0.0022 -0.0005 0.1238 0.0001 0.0104

(-10.8151) (-1.8953) (-1.3410) (1.7980) (0.2702) (6.2673)

Panel D: July 1967 to March 1982r = a0 + a1r¡4:¡1 + a2r¡52:¡5 + a3r¡156:¡53 + a4 ¹V + a5s+ a6g

Period a1 a2 a3 a4 a5 a6All -0.0552 -0.0005 -0.0013 -0.0143 -0.0007 0.0046

(-31.6943) (-0.9578) (-5.7743) (-0.4054) (-5.2407) (6.1793)Jan -0.0631 -0.0005 -0.0045 -0.1711 -0.0038 -0.0123

(-7.9314) (-0.2847) (-4.5862) (-1.7864) (-8.4704) (-4.0505)Feb-Nov -0.0532 -0.0004 -0.0011 -0.0231 -0.0004 0.0058

(-29.2124) (-0.6562) (-4.2579) (-0.5866) (-3.0217) (7.5394)Dec -0.0666 -0.0016 -0.0007 0.2267 0.0001 0.0102

(-10.9771) (-0.8759) (-1.2674) (1.9665) (0.2758) (4.2340)

Panel E: April 1982 to December 1996r = a0 + a1r¡4:¡1 + a2r¡52:¡5 + a3r¡156:¡53 + a4 ¹V + a5s+ a6g

Period a1 a2 a3 a4 a5 a6All -0.0297 0.0000 -0.0001 -0.0233 -0.0002 0.0035

(-20.3628) (0.1063) (-0.6985) (-1.2045) (-1.8569) (4.8216)Jan -0.0401 0.0004 -0.0004 0.0540 -0.0013 -0.0110

(-8.9945) (0.2767) (-0.4923) (0.6699) (-3.6897) (-3.0077)Feb-Nov -0.0284 0.0003 -0.0001 -0.0350 -0.0001 0.0042

(-18.1436) (0.6909) (-0.4204) (-1.7047) (-0.8256) (5.7574)Dec -0.0325 -0.0028 -0.0003 0.0177 0.0000 0.0106

(-5.1506) (-1.9620) (-0.5609) (0.2447) (0.0839) (4.6193)

32

Table IIIAlternative Explanations

This table investigates alternative explanations for the signi¯cance of the coe±cient on the capital gains

regressor. For Panels A and B, ¹g is calculated from a reference price using¹V jt , ¯rm j's average weekly

turnover from weeks t ¡ 52 to t ¡ 1 in the formula for the gain variable used in week t's cross-sectionalregression. Panel A replicates Panel C of Table II, replacing our original capital gains variable by ¹g. In PanelB, the relative signi¯cance of the two gain variables are compared by including both as regressors. PanelsC and D investigate whether signi¯cance was generated by the capital gains variable being correlated withsome interaction between past returns and volume over several horizons. Panels C and D add three turnoverand past return interaction terms without and with our original capital gains variable, respectively. Theparameter estimates and t-statistics (in parentheses) are obtained from the time series of the correspondingcross-sectional regression coe±cients. There are a total of 1539 weekly regressions.

Panel Ar = a0 + a1r¡4:¡1 + a2r¡52:¡5 + a3r¡156:¡53 + a4 ¹V + a5s+ a6¹g

Period a1 a2 a3 a4 a5 a6All -0.0419 -0.0003 -0.0008 -0.0160 -0.0003 0.0043

(-35.3749) (-0.9434) (-5.6612) (-0.8074) (-4.3955) (8.0694)Jan -0.0511 -0.0004 -0.0026 -0.0553 -0.0030 -0.0097

(-10.8551) (-0.3277) (-4.0810) (-0.8509) (-9.4107) (-3.9209)Feb-Nov -0.0403 -0.0001 -0.0007 -0.0266 -0.0002 0.0051

(-32.0373) (-0.3395) (-4.2724) (-1.2182) (-1.8236) (9.2848)Dec -0.0488 -0.0019 -0.0005 0.1250 0.0003 0.0103

(-10.7502) (-1.7329) (-1.3532) (1.8159) (1.2605) (6.0802)

Panel B

r = a0 + a1r¡4:¡1 + a2r¡52:¡5 + a3r¡156:¡53 + a4 ¹V + a5s+ a6g + a7¹g

Period a1 a2 a3 a4 a5 a6 a7All -0.0424 -0.0003 -0.0008 -0.0133 -0.0003 0.0028 0.0014

(-34.7482) (-1.0308) (-5.1146) (-0.6459) (-3.8746) (2.4381) (1.2590)Jan -0.0524 -0.0010 -0.0029 -0.0193 -0.0022 -0.0238 0.0154

(-10.9145) (-0.7809) (-4.2183) (-0.2889) (-7.7160) (-4.0727) (2.5766)Feb-Nov -0.0405 -0.0001 -0.0006 -0.0260 -0.0001 0.0042 0.0007

(-31.3421) (-0.2966) (-3.6772) (-1.1447) (-1.8369) (3.6161) (0.5650)Dec -0.0513 -0.0020 -0.0004 0.1160 0.0002 0.0152 -0.0048

(-10.7503) (-1.6643) (-0.9656) (1.5729) (0.9717) (4.5511) (-1.4017)

33

Panel C

r = a0 + a1r¡4:¡1 + a2r¡52:¡5 + a3r¡156:¡53 + a4 ¹V + a5 ¹V ¤ r¡4:¡1 + a6 ¹V ¤ r¡52:¡5 + a7 ¹V ¤ r¡156:¡53 + a8s

Period a1 a2 a3 a4 a5 a6 a7 a8All -0.0601 0.0013 -0.0007 -0.0805 1.2308 0.0169 0.0178 -0.0004

(-39.0925) (2.7829) (-2.9421) (-3.1500) (13.2851) (0.8890) (1.8128) (-4.4309)Jan -0.0863 -0.0115 -0.0086 -0.0261 1.6915 0.2757 0.1494 -0.0041

(-10.3072) (-4.7567) (-6.3218) (-0.3156) (4.8222) (2.8754) (3.0648) (-11.3921)Feb-Nov -0.0569 0.0020 -0.0002 -0.1055 1.1440 0.0074 0.0132 -0.0001

(-36.9647) (4.3620) (-0.8066) (-3.7325) (11.3819) (0.3675) (1.2977) (-1.5696)Dec -0.0647 0.0067 0.0023 0.1080 1.6171 -0.1455 -0.0672 0.0008

(-12.7619) (4.6382) (3.5868) (1.2784) (5.1047) (-3.2586) (-2.1602) (3.1884)

Panel D

r = a0+a1r¡4:¡1+a2r¡52:¡5+a3r¡156:¡53+a4 ¹V +a5 ¹V ¤r¡4:¡1+a6 ¹V ¤r¡52:¡5+a7 ¹V ¤r¡156:¡53+a8s+a9g

Period a1 a2 a3 a4 a5 a6 a7 a8 a9All -0.0505 -0.0004 -0.0010 -0.0448 0.8291 0.0172 0.0203 -0.0004 0.0041

(-36.9999) (-1.0135) (-5.2744) (-1.8042) (9.8004) (0.9803) (2.0141) (-5.4115) (7.7062)Jan -0.0631 0.0004 -0.0020 -0.0606 1.2519 0.0256 -0.0324 -0.0026 -0.0127

(-10.9625) (0.2253) (-2.1356) (-0.7509) (4.2349) (0.3194) (-0.7282) (-8.6123) (-5.2671)Feb-Nov -0.0481 -0.0004 -0.0010 -0.0668 0.7449 0.0230 0.0295 -0.0002 0.0052

(-33.5092) (-0.8019) (-4.7637) (-2.4386) (8.0398) (1.2122) (2.7375) (-3.0289) (9.5347)Dec -0.0619 -0.0017 -0.0004 0.1836 1.2275 -0.0471 -0.0171 0.0001 0.0102

(-12.0063) (-1.3278) (-0.8323) (2.2080) (4.2620) (-1.0533) (-0.5460) (0.3912) (5.9292)

34

Figure 1Momentum for Di®erent Past Return Horizons

This ¯gure plots the expected price change over next month in a simulated economy for momentum strategies

that sort stocks on past returns over di®erent horizons. First, 100,000 paths of fundamental values F

over 60 months are simulated according to a random walk model, assuming initial F = R = 1. The

reference price, the market price and agents' equilibrium holdings are calculated along each path, assuming

Rt+1 = VtPt + (1 ¡ Vt)Rt, where Vt is turnover ratio. At month 60, a momentum porfolio is formed that

buys the top decile (winners) and shorts the lowest decile (losers) sorted according to the change in the

fundamental value over past n months, for n = 1; 2; : : : ; 36. The expected price change over the next month

of this winner minus loser portfolio is plotted against the past return horizon used to identify winners and

losers. The graph is generated with following model parameters: annual volatility of F is ¾ = 30%; b constant

and consistent with type-r investors' absolute risk aversion coe±cient of ° = 2 (a myopic exponential utility

function); ¹ = 1=3, implying that 1/3 of investors are subject to the disposition e®ect and ¸ = 1.

0 6 12 18 24 30 361.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65

Past Return Horizon (months)

Exp

ecte

d Pr

ice

Cha

nge

over

Nex

t Mon

th (

%)

35

Figure 2Time Series of Cross-Sectional Percentiles of the Capital Gains Regressor

This ¯gure plots the time series of the empirical 10th, 50th and 90th percentiles of the cross-sectionaldistribution of the capital gains regressor. The sample period is from July 1967 to December 1996, for atotal of 1539 weeks. Each week, we include all stocks (with sharecode 10 or 11) listed on NYSE and AMEXwhich have at least ¯ve years of historical tading data from mini-CRSP. The previous ¯ve years of returnand turnover data are used to calculate the capital gains variable as one less the ratio of the beginning ofweek t¡ 1 reference price to the end of week t¡ 2 price, where the week t¡ 1 reference price is the averagecost basis obtained from the formula

Rt¡1 =1

k

260Xn=1

ÃVt¡1¡n

n¡1Y¿=1

[1¡ Vt¡1¡n+¿ ]!Pt¡1¡n

with k a constant that makes the weights on past prices sum to one.

Jul67 Dec70 Jun74 Nov77 Apr81 Oct84 Mar88 Sep91 Feb95−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

Date

Cap

ital g

ains

ove

rhan

g

10 percentilemedian90 percentile

36

Appendix

Equilibrium and Demand Functions in a 3-Date ExponentialUtility Model

Within the context of the model described in Section I, we consider the date 0 valuation of

two securities: a risk-free asset with a return of 0 in in¯nitiely elastic supply and a risky

stock that pays a liquidating dividend ~F2 at date 2. Trading at date 1 occurs after receiving

a normally distributed signal about F2 that resolves half of the uncertainty about the ¯nal

payo®. That is,

E1[F2] = E0[F2] + ²1; ²1 » N(0; ¾2)and Var1(F2) = ¾

2. It follows that

Var0(F2) = E0[Var1(F2)] + Var0(E1[F2]) = 2¾2

There are two types of price-taking investors: type-r, whose demand function has weight

1 ¡ ¹, has CARA utility over terminal wealth without intermediate consumption. This

type chooses date 0 risky asset shareholding D0 and date 1 shareholding D1 in the stock to

maximize expected utility of ¯nal wealth:

MaxD0;D1 E[¡e¡°W2 ]

where ° is his absolute risk aversion coe±cient. Denote Drt (Pt) as type r's optimal demand

at date t given price Pt, and de¯ne Ft as the price at which he would optimally hold one

share of the stock, assuming knowledge of how type-d investors in°uence future equilibrium

prices. The latter investor-type, with weight ¹ in the economy has date t demand given by

Drt (Pt) + ¸bt(Rt ¡ Pt)

where

bt =Drt (Pt)¡Dr

t (Ft)

Ft ¡ Ptwith variables and their dynamics de¯ned in the body of the paper.

We calculate the type-r investors optimal demand function (in shares) for the stock,

given the conjecture that the linear equilibrium price function

Pt = wFt + (1¡ w)Rt;where w = 1

1 + ¹¸

applies. By the argument in the body of the text, the market will clear at the above

conjectured price as long as type r's demand exists and is ¯nite.

We now explicitly calculate rational agent's demand Drt (Pt) by backwards induction, and

show that b0 and b1 are positive. At date 2, the risky asset's value is F2 as there is only a

37

liquidating payout. Hence at date 1, the rational agent's optimal demand at any price P1will be the same as that in a fully rational economy. With CARA utility and a normally

distributed date 2 payo®, the date 1 demand of type-r investors must satisfy:

D1(P1) =E1[F2]¡ P1

°¾2=µ + ²1 ¡ P1

°¾2

which generates a fully rational equilibrium price satisfying

F1 = µ + ²1 ¡ °¾2 (27)

The last two equations imply that

b1 =D1(P1)¡ 1F1 ¡ P1 =

1

°¾2> 0

We obtain the date 1 indirect utility function J1(W1; P1) by evaluating the expected utility

at the optimal demand above:

J1(W1; P1) = ¡e¡°µW1+

(E1[F2]¡P1)22°¾2

¶

Next, we turn to rational agent's optimal demand at date 0. At any price P0 at date 0,

the type-r investors choose D0 = D0(P0) to

Maximize E0[J1(W1; P1)]

where P1 = wF1 + (1¡ w)R1 is the equilibrium price at date 1, and

W1 =W0 +D0(P1 ¡ P0) =W0 +D0(wF1 + (1¡ w)R1 ¡ P0)

which is equivalent to

MaxD0 ¡ E0"e¡°D0(wF1+(1¡w)R1¡P0)¡ (µ+²1¡wF1¡(1¡w)R1)2

2¾21

#(28)

Upon substituting (27) and collecting terms, equation (28) can be rewritten as

MaxD0 ¡ E0he¡(m0+m1²1+m2²21)

ior equivalently, MaxD0 ¡ e¡(m0¡ m21

4m2)E0

·e¡m2

³²1+

m12m2

´2¸(29)

where m0;m1 and m2 are deterministic functions of model parameters, price P0, reference

price R1 = ºP0 + (1 ¡ º)R0 (which is known at date 0 given R0 and an exogenous º), and

38

shares invested in the stock D0, as follows:

m0(D0) = °h1D0 +h222¾2

m1(D0) =h2(1¡ w)

¾2+ °wD0

m2 =(1¡ w)22¾2

where h1 and h2 are functions of P0

h1 = w(µ ¡ °¾2) + (1¡ w)R1 ¡ P0h2 = (1¡ w)(µ ¡R1) + w°¾2

Denote y = ²1 +m1

2m2, then y » N( m1

2m2; ¾2). Note that the probability density function

of the normal random variable y is e¡ 12¾2

³y¡ m1

2m2

´2. This is of the same functional form as

e¡m2y2, and greatly simpli¯es the computation of the expectation E[e¡m2y2 ]. Note that the

expression to be maximized here involves only the expectation of an exponential of a normal

random variable ²1 and its square.

We now make use of moment generating functions of normal and Â2 random variables

to derive the solution. By completing squares and using the fact that the integral of a

probability density function is 1, we get

E[e¡m2y2] =1p

1 + 2m2¾2e¡ m214m2(1+2m2¾

2)

Substituting this into (29), type r investors solve the following problem:

MaxD0 ¡1p

1 + 2m2¾2e¡(m0¡ m21

4m2)e¡ m214m2(1+2m2¾

2)

or equivalently, MaxD0 (m0 ¡ m21

4m2

) +m21

4m2(1 + 2m2¾2)

or equivalently, MaxD0 °h1D0 +h222¾2

¡ m21¾2

2(1 + 2m2¾2)

Since m1 is linear in D0, the function being optimized above is quadratic in D0. The

coe±cient for theD20 term is negative, and hence a maximum exists. The ¯rst order condition

implies that the optimal choice D0(P0) satis¯es

(°w2¾2)D0(P0) = h1(P0) + (1¡ w) ((1¡ w)h1(P0)¡ wh2(P0))For a given price P0, the type-r investor's optimal demand at date 0 is

D0(P0) =1

°w2¾2(h1(P0) + (1¡ w) ((1¡ w)h1(P0)¡ wh2(P0)) (30)

39

By de¯nition, F0 satis¯es

°w2¾2 = h1(F0) + (1¡ w) ((1¡ w)h1(F0)¡ wh2(F0))

or 1 =1

°w2¾2(h1(F0) + (1¡ w) ((1¡ w)h1(F0)¡ wh2(F0)) (31)

Subtracting (31) from (30),

D0(P0)¡ 1 = 1

°w2¾2¡(1 + (1¡ w)2)(h1(P0)¡ h1(F0)) + (1¡ w)w(h2(F0)¡ h2(P0))

¢(32)

Upon substituting h1; h2, and using R1 = ºP0 + (1¡ º)R0,

h1(P0)¡ h1(F0) = (1¡ º(1¡ w))(F0 ¡ P0)h2(F0)¡ h2(P0) = (1¡ w)º(P0 ¡ F0)

Plugging these back into (32),

D0(P0)¡ 1 = 1

°w2¾2¡(1 + (1¡ w)2)(1¡ (1¡ w)º)¡ (1¡ w)2wº¢ (F0 ¡ P0) (33)

implying

b0 =1

°w2¾2¡(1 + (1¡ w)2)(1¡ (1¡ w)º)¡ (1¡ w)2wº¢ > 0

40

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